A matter of degree
Statistical evidence for causality combines observed data with a mathematical model of the world
Causal evidence varies in terms of complexity of math/assumptions: a matter of degree
Model-based inferences about causality depend on complex statistical models with many assumptions
Design-based inferences about causality use carefully controlled comparisons with simple, transparent models and assumptions
Whatever our approach…
do the assumptions needed to use this mathematical tool reasonably fit reality?
In 1956, the state of Connecticut responded to high rates of automobile fatalities by imposing harsher penalties for speeding.
What would a conditioning strategy look like?
What confounding does this address?
What assumptions do we make?
What confounding does this address?
What assumptions do we make?
What kinds of omitted variables does this comparison address?
What assumptions do we make?: none of the following
A good example:
We can combine before-and-after with conditioning:
Extend the interrupted time-series:
This can reveal the unbiased causal effect on treated, assuming “parallel trends” - That is to say: “control” cases are not counterfactuals for “treated” case, but are the counterfactual trend
If parallel trends assumption holds, what kinds of confounding does this design eliminate?
Implementation:
Example: Card and Krueger (2000)
Do increases in the minimum wage increase unemployment in fast food?
Two ways to use regression:
\[Y_it = \beta_0 + \beta_1 D_i + \beta_2 T_{t} + \beta_3 D_i \times T_{t} + \epsilon_{it}\]
Where \(D_i\) is an indicator for being a unit that is ever treated (1 yes, 0 no). \(T_t\) is an indicator for the observation being after the treatment takes place (1 if yes, 0 if no).
\[Y_{post} - Y_{pre} = \beta_0 + \beta_1 D_i + \epsilon_i\]
How do we validate the parallel trends assumption?
Placebo Tests:
If we have multiple time-periods, multiple cases, we get generalized difference-in-differences:
\[Y_{it} = \alpha_i + \alpha_t + \beta D_{i}T_{t} + \epsilon_{it}\]
We can add…
…to any model. It removes any confounding from variables \(Z\) that:
BUT, despite apparent similarity, TWFE \(\neq\) DID
Example:
Example:
Example:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.5914434 0.69640003 3.721199 0.001563083
## x 0.1732158 0.05813435 2.979578 0.008034175
Example:
In this case, the overall relationship between \(X\) and \(Y\) is positive. But within each unit (\(a \dots e\)), the relationship is negative!
Unobserved factors might make each unit have higher levels of x and higher levels of y, but as x increases within a unit (where these unobserved factors are constant), y decreases.
fixed effects allow us to extract this within unit relationship between \(X\) and \(Y\)
Contrast to pooled effects where we compare all observations to each other, not accounting for any unit-specific effects
How to use fixed effects
We add dummy variables for each unit
What does this do?
Fixed effects results:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.178014 0.12312328 42.05552 3.872839e-16
## x -1.063517 0.03482452 -30.53932 3.262312e-14
## gb 5.461419 0.18591208 29.37635 5.574230e-14
## gc 10.295492 0.30459018 33.80113 8.021837e-15
## gd 15.614313 0.43565465 35.84103 3.562884e-15
## ge 20.624415 0.57063358 36.14301 3.171759e-15
Staggered Treatment: Bacon-Goodman 2021
Multiple Treatments: https://arxiv.org/pdf/1803.08807.pdf
Continuous Treatment: https://psantanna.com/files/Callaway_Goodman-Bacon_SantAnna_2021.pdf
What was the effect of enlistment in the US Civil War on voting for the Republican party?
\[GOP_{ie} = \alpha_i + \alpha_e + \beta Enlist_i \times PostWar_e + \epsilon_y + \epsilon_i\]
dd1 = felm(gop_vs ~ postbellum:vet_alt_pct | #interaction of enlistment rate, post-war
target_fips + state_election | #county, state-election fixed effects
0 | #no Instrumental variable
target_fips + election_year , # cluster errors by county, election year
merged_use)
A placebo test/relaxed assumptions
\[GOP_{ie} = \alpha_{i} + \alpha_{e} + \sum_{y = 1854}^{1920} \beta_y EnlistmentRate_i * Year_y + \epsilon_i + \epsilon_y\]
year_levels = c(1860, seq(1854,1858,2), seq(1862,1920,2)) %>% as.character()
merged_panel[, year_f := factor(election_year, levels = year_levels)]
g1 = felm(gop_vs ~ 1 + year_f*vet_alt_pct | #interaction of YEAR and enlistment-rate
target_fips + state_election | #county and state-election FE
0 | #no IV
target_fips + election_year ,#cluster by county/ election year
merged_use)
Key Assumptions:
Caveats:
Address confounding / omitted variable bias in a different way:
Distinguishing “natural experiment” from experiments:
An observational study where causal inference comes from the design that draws on randomization.
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Two approaches:
Decisions:
Assumptions:
Follows from Wald estimator for non-compliance:
Decisions
Assumptions: